Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 4 / d

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- Solution d

Solution

The tensor product of the defining representation and its dual is the endomorphism representation. Splitting an endomorphism into its matrix trace and traceless part gives

d_{1} \otimes d_{2} ≅ 4 \otimes \overline{4} = 1 \oplus 15 .

In Dynkin labels, the two irreducible summands have highest weights

[0, 0, 0]

and

[1, 0, 1]

.

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